BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Luis Dieulefait (Universitat de Barcelona) DTSTART:20201215T123000Z DTEND:20201215T133000Z DTSTAMP:20260416T005027Z UID:NTRV/1 DESCRIPTION:Title: Pot entially diagonalizable modular lifts of arbitrarily large weight\nby Luis Dieulefait (Universitat de Barcelona) as part of Number Theory and Re presentations in Valparaiso\n\n\nAbstract\nI will begin this talk by recal ling the notion of "potential diagonalizability"\, due to Barnet-Lamb\, Ge e\, Geraghty and Taylor\, and how (and why) this notion appears in Automor phy Lifting Theorems (in the work of the aforementioned authors).\nThen I will present the main result of this talk\, which is joint work with Iván Blanco: existence of modular lifts of arbitrarily large weight (of a give n residual modular Galois representations) which are potentially diagonali zable. In the non-ordinary case\, the proof of this result requires a comb ination of local and global results for Galois deformation rings\, a local to global principle due to Böckle and a potential variant of the result we want to prove due to Barnet-Lamb\, Gee\, Geraghty and Taylor.\nI will a lso explain how this result is a useful tool in the proof of some cases of Langlands functoriality.\n LOCATION:https://researchseminars.org/talk/NTRV/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Jessica Fintzen (IAS\, Cambridge University\, Duke University) DTSTART:20201215T140000Z DTEND:20201215T150000Z DTSTAMP:20260416T005027Z UID:NTRV/2 DESCRIPTION:Title: Rep resentations of $p$-adic groups and applications\nby Jessica Fintzen ( IAS\, Cambridge University\, Duke University) as part of Number Theory and Representations in Valparaiso\n\n\nAbstract\nThe Langlands program is a f ar-reaching collection of conjectures that relate different areas of mathe matics including number theory and representation theory. A fundamental pr oblem on the representation theory side of the Langlands program is the co nstruction of all (irreducible\, smooth\, complex) representations of $p$- adic groups. I will provide an overview of our understanding of the repres entations of $p$-adic groups\, with an emphasis on recent progress.\n\nI w ill also outline how new results about the representation theory of $p$-ad ic groups can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at $p$\, which is join t work with Sug Woo Shin. This simplifies earlier constructions of attachi ng Galois representations to automorphic representations\, i.e. the global Langlands correspondence\, for general linear groups. Moreover\, our resu lts apply to general $p$-adic groups and have therefore the potential to b ecome widely applicable beyond the case of the general linear group.\n LOCATION:https://researchseminars.org/talk/NTRV/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Haruzo Hida (University of California\, Los Angeles) DTSTART:20201215T170000Z DTEND:20201215T180000Z DTSTAMP:20260416T005027Z UID:NTRV/3 DESCRIPTION:Title: $p$ -Local indecomposability of non-CM modular $p$-adic Galois representations and the structure of Hecke algebras\nby Haruzo Hida (University of Ca lifornia\, Los Angeles) as part of Number Theory and Representations in Va lparaiso\n\n\nAbstract\nA conjecture by R. Greenberg asserts that a modula r 2-dimensional $p$-adic Galois representation of a cusp form of weight la rger than or equal to 2 \nis indecomposable over the $p$-inertia group u nless it is induced from an imaginary quadratic field. \nI start with a s urvey of the known results and try to reach a brief description of new cas es of indecomposability.\n LOCATION:https://researchseminars.org/talk/NTRV/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Ariel Pacetti (Universidad Nacional de Córdoba) DTSTART:20201216T123000Z DTEND:20201216T133000Z DTSTAMP:20260416T005027Z UID:NTRV/4 DESCRIPTION:Title: On the number of Galois orbits of newforms\nby Ariel Pacetti (Universidad Nacional de Córdoba) as part of Number Theory and Representations in Val paraiso\n\n\nAbstract\nA conjecture of Maeda states that all newforms of l evel $1$ and weight $k \\ge 16$ are Galois conjugate. This striking observ ation has very deep implications in the study of Galois representations an d until now stays as a completely open problem. A natural question is to u nderstand what happens when we move from level $1$ to modular forms of lev el $\\Gamma_0(N)$\, with $N > 1$. In this talk we will present a list of i nvariants of Galois orbits of newforms of level $N$\, giving a lower bound for the number of Galois orbits (when $k$ is large enough). It is a natur al question to understand whether such a lower bound is attached (a questi on related to Maeda's conjecture).\n LOCATION:https://researchseminars.org/talk/NTRV/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Gonzalo Tornaría (Universidad de la República) DTSTART:20201216T140000Z DTEND:20201216T150000Z DTSTAMP:20260416T005027Z UID:NTRV/5 DESCRIPTION:Title: An explicit Waldspurger formula for Hilbert modular forms\nby Gonzalo Tor naría (Universidad de la República) as part of Number Theory and Represe ntations in Valparaiso\n\n\nAbstract\nComputing central values of $L$-func tions attached to modular forms is interesting because of the arithmetic i nformation they encode. These values are related to Fourier coefficients o f half-integral weight modular forms and the Shimura correspondence\, as s hown in great generality by Waldspurger.\n\nWhen $g$ is a classical modula r form of odd square-free level\, a theorem of Baruch and Mao shows the ex istence of a finite set of modular forms of half-integral weight\, togethe r with an explicit formula for twisted central values $L(1/2\,g\,D)$ for e very fundamental discriminant $D$.\n\nIn this talk I will present a genera lization of this result to all levels except perfect squares\, and to Hilb ert modular forms over an arbitrary totally real number field.\n\nJoint wo rk with Nicolás Sirolli (Universidad de Buenos Aires).\n LOCATION:https://researchseminars.org/talk/NTRV/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Giancarlo Lucchini Arteche (Universidad de Chile) DTSTART:20201216T170000Z DTEND:20201216T180000Z DTSTAMP:20260416T005027Z UID:NTRV/6 DESCRIPTION:Title: Loc al-global principles for homogeneous spaces over some geometric global fie lds\nby Giancarlo Lucchini Arteche (Universidad de Chile) as part of N umber Theory and Representations in Valparaiso\n\n\nAbstract\nLocal-global principles are a classical type of problem in Number Theory\, both over n umber fields and over global fields in positive characteristic. Concerning the local-global principle for the existence of rational points\, there i s a classic obstruction known as the Brauer-Manin obstruction\, which is c onjectured to explain all failures of this principle for homogeneous space s of connected linear groups.\nIn the last few years\, there has been an i ncreasing interest in fields of a more ``geometric" nature that are amenab le to local-global principles as well. These include\, for instance\, func tion fields of curves over discretely valued fields\, by analogy with the positive characteristic case. It is in this context that I will present re cent work with Diego Izquierdo on local-global principles for homogeneous spaces with connected stabilizers. We will see that\, although some of the known results for number fields have direct analogs (that can be obtained in the same way)\, the particularities of these new fields bring up new c ounterexamples that cannot be explained by the Brauer-Manin obstruction\, contrary to the number field case.\n LOCATION:https://researchseminars.org/talk/NTRV/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Guy Henniart (Université Paris-Saclay\, Orsay) DTSTART:20201217T123000Z DTEND:20201217T133000Z DTSTAMP:20260416T005027Z UID:NTRV/7 DESCRIPTION:Title: On stability for exterior and symmetric square for ${\\rm GL}(n)$\nby Guy Henniart (Université Paris-Saclay\, Orsay) as part of Number Theory and Representations in Valparaiso\n\n\nAbstract\nIn local to global arguments\ , it is very useful to know the behaviour of $L$ and epsilon factors under very ramified twists. For GL$(n\, F)$\, $F$ local non-archimedean\, Cogde ll\, Shahidi and Tsai proved that if one twists a given cuspidal represent ation $\\pi$ with a ramified enough character\, then the $L$ and epsilon f actors for the exterior and symmetric square acquire a simple shape. They establish that ``stability" property\, mainly by a local\, intricate proof . We propose a global to local proof\, taking advantage of progress in the global Langlands correspondence for GL$(n)$ over number fields.\n LOCATION:https://researchseminars.org/talk/NTRV/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Sandeep Varma (Tata Institute of Fundamental Research) DTSTART:20201217T140000Z DTEND:20201217T150000Z DTSTAMP:20260416T005027Z UID:NTRV/8 DESCRIPTION:Title: On residues of certain intertwinig operators\nby Sandeep Varma (Tata Inst itute of Fundamental Research) as part of Number Theory and Representation s in Valparaiso\n\n\nAbstract\nLet $\\rm G$ be a connected reductive group over a finite extension\n$F$ of $\\mathbb{Q}_p$. Let $\\rm P = M N$ be a Levi decomposition of\na maximal parabolic subgroup of $\\rm G$\, and $\\s igma$ an irreducible unitary supercuspidal representation of ${\\rm M}(F)$ . One can then consider the representation $\\text{Ind}_{{\\rm P}(F)}^{{\\ rm G}(F)} \\sigma$ (normalized parabolic induction). This induced represen tation is known to be either irreducible or of\nlength two. The question o f when it is irreducible turns out to be (conjecturally) related to local $L$-functions\, and also to poles of a family of so called intertwining op erators. \n\nBecause of this\, one would like to:\n\n(a) get expressions d escribing residues of certain families of intertwining operators\; and\n\n (b) interpret these residues suitably\, using the theory of\nendoscopy whe n applicable.\n\nThere is an approach pioneered by Freydoon Shahidi to imp lement such a programme\, which was developed further by him as well as by David Goldberg\, Steven Spallone\, Wen-Wei Li and Xiaoxiang Yu\, in sever al cases (i.e.\, for various choices of $\\rm G$ and $\\rm P$). We will di scuss (a) in the cases where $\\rm G$ is an almost simple group whose abso lute root system is of exceptional type or of types $B_n$ or $D_n$ with $n > 3$\, and where $\\rm P$ is a `Heisenberg parabolic subgroup'. If time p ermits\, partial results towards and speculations concerning (b) will be d iscussed.\n LOCATION:https://researchseminars.org/talk/NTRV/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Vincent Sécherre (Université de Versailles) DTSTART:20201217T170000Z DTEND:20201217T180000Z DTSTAMP:20260416T005027Z UID:NTRV/9 DESCRIPTION:Title: Sel fdual cuspidal representations of ${\\rm GL}(N)$ and distinction by an inn er involution\nby Vincent Sécherre (Université de Versailles) as par t of Number Theory and Representations in Valparaiso\n\n\nAbstract\nLet $n $ be a positive integer\, $F$ be a non-Archimedean locally compact field o f odd residue characteristic $p$ and $G$ be an inner form of GL$(2n\,F)$. This is a group of the form GL$(r\,D)$ for a positive integer $r$ and divi sion $F$-algebra $D$ of reduced degree $d$ such that $rd=2n$. Let $K$ be a quadratic extension of $F$ in the algebra of matrices of size $r$ with co efficients in $D$\, and $H$ be its centralizer in $G$. We study selfdual c uspidal representations of $G$ and their distinction by $H$\, that is\, th e existence of a nonzero $H$-invariant linear form on such representations . When $F$ has characteristic 0\, we characterize distinction by $H$ for c uspidal representations of $G$ in terms of their Langlands parameter\, pro ving in this case a conjecture by Prasad and Takloo-Bighash.\n LOCATION:https://researchseminars.org/talk/NTRV/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Ramla Abdellatif (Université de Picardie Jules Verne) DTSTART:20201218T123000Z DTEND:20201218T133000Z DTSTAMP:20260416T005027Z UID:NTRV/10 DESCRIPTION:Title: Re striction of $p$-modular representations of $p$-adic groups to minimal par abolic subgroups\nby Ramla Abdellatif (Université de Picardie Jules V erne) as part of Number Theory and Representations in Valparaiso\n\n\nAbst ract\nGiven a prime integer $p$\, a non-archimedean local field $F$ of res idual characteristic $p$ and a standard Borel subgroup $P$ of $GL_{2}(F)$\ , Pa${\\check{\\text{s}}}$k$\\overline{\\text{u}}$nas proved that the rest riction to $P$ of (irreducible) smooth representations of $GL_{2}(F)$ over $\\overline{\\mathbb{F}}_{p}$ encodes a lot of information about the full representation of $GL_{2}(F)$ and that it leads to useful statement about $p$-adic representations of $GL_{2}(F)$. Nevertheless\, the methods used at that time by Pa${\\check{\\text{s}}}$k$\\overline{\\text{u}}$nas heavil y relied on the understanding of the action of certain spherical Hecke ope rator and on some combinatorics specific to the $GL_{2}(F)$ case. This met hod can be transposed case by case to for some other quasi-split groups of rank $1$\, but this is not very satisfying as such. \n\nThis talk will re port on a joint work with J. Hauseux. Using Emerton's ordinary parts funct or\, we get a more uniform context which sheds a new light on Pa${\\check{ \\text{s}}}$k$\\overline{\\text{u}}$nas' results and allows us to generali ze very naturally these results for arbitrary rank $1$ groups. In particul ar\, we prove that for such groups\, the restriction of supersingular repr esentations to a minimal parabolic subgroup is always irreducible.\n LOCATION:https://researchseminars.org/talk/NTRV/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Barrera Salazar (Universidad de Santiago de Chile) DTSTART:20201218T140000Z DTEND:20201218T150000Z DTSTAMP:20260416T005027Z UID:NTRV/11 DESCRIPTION:Title: $p $-adic variation via evaluations on the cohomology\nby Daniel Barrera Salazar (Universidad de Santiago de Chile) as part of Number Theory and Re presentations in Valparaiso\n\n\nAbstract\nI will explain a method to cons truct automorphic $p$-adic $L$-functions and to study the geometry of eige nvarieties. This is based on the construction and study of certain linear 'evaluation' functionals on the cohomology of the relevant arithmetic mani fold. We will give more details in the context of the reductive group $GL_ {2n}$ where this methods was applied successfully. This talk is based on j oint work with C. Williams and joint work with M. Dimitrov and C. Williams .\n LOCATION:https://researchseminars.org/talk/NTRV/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Anne-Marie Aubert (CNRS\, Sorbonne Université) DTSTART:20201218T170000Z DTEND:20201218T180000Z DTSTAMP:20260416T005027Z UID:NTRV/12 DESCRIPTION:Title: $C ^*$-blocks and crossed products for classical $p$-adic groups\nby Anne -Marie Aubert (CNRS\, Sorbonne Université) as part of Number Theory and R epresentations in Valparaiso\n\n\nAbstract\nLet $G$ be the group of $F$-po ints of a quasi-split reductive connected group over a local field $F$. Fo r $F$ real\, Wassermann proved in 1987\, by noncommutative-geometric metho ds\, that each connected component of the tempered dual of $G$ has a simpl e and beautiful geometric structure that encodes the reducibility of induc ed representations. For $F$ $p$-adic\, the existence of such a structure i s far from straightforward. It was established for certain particular conn ected components by R. Plymen and his students.\n\nWe will present a joint work with Alexandre Afgoustidis which first provides a necessary and suff icient condition\, in terms of the Knapp-Stein-Silberger R-groups\, for th e existence of a Wassermann type theorem\, and secondly\ndetermine explici tly the components for which this condition is satisfied when $G$ is a cla ssical $p$-adic group.\n LOCATION:https://researchseminars.org/talk/NTRV/12/ END:VEVENT END:VCALENDAR